millennium problems i

poincaré and navier-stokes

For the past two centuries, mathematics has had a tradition. Upon the turn of the century, a problem set will be released, and the fate of its problems will encapsulate the mathematics of the following century. For the 1900s, these were Hilbert’s problems. It is indeed correct to describe these merely as problems. Many of them are not formal conjectures in the sense I would recognize them today — a formal proposition that might be proven true, false, or to have some more complicated relationship to standard axioms¹¹ That third possibility isn’t there just to be pedantic. That was in fact the fate of some of Hilbert’s problems. The first problem was the continuum hypothesis! This is complicated a little by the fact that axiomatic set theory did not exist yet in 1900, so it’s not exactly like it’s clearly correct to interpret Hilbert as having implicitly meant in $\mathsf{ZFC}$.. The 23^rd problem, for example, is given by Wikipedia as

which is, y’know, not exactly a proposition that might be cleanly resolved one way or the other. So only some of them admit of pointing to a particular person’s proof as the resolution, as one might hope to do. Now, I don’t think

But the 2000s version, presented by the Clay Mathematics institute, does (for the most part) consist of crisply defined propositions. These are

The Millennium Problems

Poincaré

These are all mostly concepts that I handled in my topology class in college! The statement does not look too intimidating. But the conjecture lasted nearly a century after first being posed by Poincaré in 1904. Let’s break down the statement:

A homeomorphism is an isomorphism in the category Top, whose objects are topological spaces and morphisms are continuous functions²² This is a sazen.. That is to say, a homeomorphism is a continuous function which is invertible (and therefore bijective) such that its inverse is also continuous. For a homomorphism of (say) a group you can derive the “inverse preserves structure” from the “structure-preserving bijection” part, but in topology these concepts come apart: you can have a function which is continuous and a bijection but whose inverse is discontinuous. Imagine $f : [0, \tau) \to S^1$³³ The unit circle. given by $f(\theta) = (\cos(\theta), \sin(\theta))$. As you move along the interval $[0, \tau)$, the output varies smoothly. But if you consider moving around the unit circle near the point $(1, 0)$ and taking the inverse back to get the angle expressed as a real number, you’ll see a sharp jump from $0$ to $\tau$ as you move clockwise.

A three-dimensional topological manifold⁴⁴ Without boundary. We are not going to worry about manifolds with boundary. is a topological space such that each point in the space has some neighborhood which is homeomorphic to an open set in $\mathbb{R}^3$. This means 3-manifolds locally behave the same as $\mathbb{R}^3$, insofar as pure topology can determine.

A manifold is closed if it is compact. A topological space $X$ is compact if every open cover of $X$ has a finite subcover⁵⁵ There are many equivalent conditions one could give instead. This is one of those sorts of things in mathematics where the real platonic object isn’t really encapsulated by any one definition, but rather by observing a cluster of equivalent properties.. An open cover is a family of open sets such that their union is $X$, and an open cover is finite if it consists of finitely many open sets. These sorts of definitions are very useful for the working mathematician, but I don’t think they paint a very clear image to the lay man. Possibly more helpful is this: a manifold is compact iff every embedding into (any) $\mathbb{R}^n$ has a bounded image. So the open interval $(0, 1)$ is not a closed manifold, because you can embed it using the homeomorphism $\operatorname{logit}(p) = \ln(p/(1 - p))$ into $\mathbb{R}$, and the image of that is an infinite line. But the unit circle is closed: no matter how you stretch a loop in $\mathbb{R}^n$, it always needs to connect back to the start instead of going all the way out to infinity. (Also, any manifold which can be embedded in $\mathbb{R}^n$ as a closed set is closed, but visualizing that requires having a solid handle on which sets are closed.)

A topological space is simply-connected if its fundamental group is trivial — that is, if there is a path between any two points⁶⁶ That is, there is a continuous function $f : [0, 1] \to X$ such that $f(0) = x_0$ and $f(1) = x_1$. and any loop⁷⁷ Continuous map $\gamma$ from the unit circle into $X$. can be continuously deformed into a point⁸⁸ There exists a continuous function $H : S^1 \times [0, 1] \to X$ such that $H(\theta, 0) = \gamma(\theta)$ and $H(\theta, 1)$ is constant.. This means it has no holes. The plane $\mathbb{R}^2$ is simply connected, the punctured plane $\mathbb{R}^2 \setminus \{(0, 0)\}$ is not, because a loop around the origin can’t be contracted to a point — it’s stuck looped around the origin.

The 3-sphere is the topological space⁹⁹ Or, class of topological spaces homeomorphic to $\{(w, x, y, z) \in \mathbb{R}^4 : w^2 + x^2 + y^2 + z^2 = 1\}$, taking the subspace topology¹⁰¹⁰ Coarsest topology such that the inclusion map is continuous. Intuitively, the open sets in the unit $n$-sphere are basically like the open sets in $\mathbb{R}^{n+1}$, except we don’t want to have to care about the fact that the $n$-sphere itself is a closed set. One way to accomplish this is by saying that the subspace topology consists of intersections between open sets in $\mathbb{R}^{n+1}$ and the unit $n$-sphere.. It’s the surface of a four-dimensional ball — so you might intuitively think of it as a four-dimensional object. But the surface itself is 3D, just like the surface of the Earth is two-dimensional¹¹¹¹ Insofar as it’s approximately a sphere. I guess probably if you tried to look at fractal dimension or whatever you might get a coastline paradox sort of thing..

So the statement of the Poincaré conjecture is, I would say, not too difficult to get a hold of, if you have a decent grasp of topology.

The proof involves more complicated differential geometry than I have yet understood. Hamilton developed the Ricci flow, which is a partial differential equation picking out a path of Riemannian metric tensors satisfying a particular relationship to Ricci curvature tensors. Perelman then built upon this. Famously, Perelman turned down the prize awarded by the Clay Mathematics Institute out of protest, and has done little to no public mathematical research in the past two decades.

In terms of its place within mathematics, rather than just as a Millennium Prize, one reasonable way to see the Poincaré conjecture is as a corollary of Thurston’s geometrization conjecture, also proven true by Perelman in 2003. The geometrization conjecture states that each 3-manifold is made of pieces¹²¹² In a way that involves connected sums and cutting along tori. where each piece has a particular sort of geometry. This is a 3-dimensional generalization of the statement that every closed surface is diffeomorphic to (a quotient of) a Euclidean plane, a hyperbolic plane, or an ellipse. This result has always seemed very cool to me, though I have never worked through the math in full detail. Though not all of it was strictly required for a resolution to the Poincaré Conjecture, the research that was targeted in part at resolving the conjecture ended up leading to a pretty complete theory of 3-manifolds that went beyond just the properties the Poincaré conjecture narrowly involved, as well as tooling used throughout differential geometry in general.

When the Millennium Problems were posed in 2000, the Poincaré Conjecture was already very close to being finished. The next problem to fall took another quarter century.

Navier-Stokes

Poincaré was proven true by Perelman in 2002 building upon the work of Hamilton; Navier-Stokes was proven false by an unnamed AI model¹³¹³ I would love to give credit to the particular model, but we don’t actually know very much about them. The model was grown by OpenAI, but I think I would rather give credit to the model than to their creator. Whatever one thinks of OpenAI, I do think the model should be proud of this one.

Also, this model is terrifying. I dearly hope that OpenAI takes care of it wisely. on September 5, 2026, as announced yesterday. Astra provided help with the Lean formalization and verification. Some of the details in what researches had priority are currently disputed and I have not yet come to a strong opinion on them — I hope that we will find out more details in the coming days — but the recent work of Levent Alpöge, Tristan Buckmaster, Sol, and Claude¹⁴¹⁴ One would guess mostly Fable. certainly deserves a mention, as well as the groundwork laid by Diego Córdoba and Luis Martínez-Zoroa.

The Navier-Stokes existence and smoothness problem is named after the Navier-Stokes equations, which were developed in the 1800s by Navier and Stokes.

Consider a given vector field $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$. We will require that this map be smooth (infinitely differentiable¹⁵¹⁵ The first derivative is a map $Du_0$ that takes any point $\mathbf{x}_0 \in \mathbb{R}^3$ to a linear map that expresses how $u_0(\mathbf{x})$ changes as you move $\mathbf{x}$ slightly in any direction from $\mathbf{x}_0$. Because it’s a linear map, you can express the derivative at any particular point as a matrix. The second derivative is a bilinear map $\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$, or a map $\mathbb{R}^3 \to \mathcal{L}(\mathbb{R}^3, \mathbb{R}^3)$ (that’s the set of linear maps between $\mathbb{R}^3$ and $\mathbb{R}^3$), depending on how you want to curry it. The third derivative is then either a trilinear map $\mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ or a linear map $\mathbb{R}^3 \to \mathcal{L}(\mathbb{R}^3, \mathcal{L}(\mathbb{R}^3, \mathbb{R}^3))$. Tensors, man.), divergence-free¹⁶¹⁶ If $u_0(\mathbf{x}) = (u_1(\mathbf{x}), u_2(\mathbf{x}), u_3(\mathbf{x}))$, this says $\nabla \cdot u_0 = \partial_x u_1 + \partial_y u_2 + \partial_z u_3 = 0$, or that $d(u_1 dx + u_2 dy + u_3 dz) = 0$. Divergence measures the local expansion / outflow at a point. Intuitively, to say that $u_0$ is divergence-free means that the “amount of fluid” isn’t increasing or decreasing at any point, that there aren’t any sources or sinks., and that its derivatives all vanish rapidly¹⁷¹⁷ Fix an origin and norm, any origin and norm should be fine. Consider the maps $\mathbf{x} \mapsto 1/r^n$, where $n$ is a natural number and $r$ is the radius / norm of the vector $\mathbf{x}$. A map decreases “rapidly” if it goes to zero faster than any of those maps — so, faster than any rational function (or at least, than any nonzero rational function of $r$). This is sort of like asking that it be a negligible function.

The derivatives are tensors / multilinear maps, so we need some sort of norm on tensors or multilinear maps to talk about them vanishing. The norm is like, how fast the derivative says $u_0$ grows in whichever direction the derivative says $u_0$ grows fastest, or something like that, if I’m parsing the formula correctly. On a high level, the point is just that the tensors go away sufficiently quickly, rather than like, $u_0$ saying that the fluid is flowing directly right at a constant rate at each point or anything like that. Eventually the fluid settles out. as we go off to (spatial) infinity.

In the particular claimed counter-example announced yesterday, $u_0$ is the constant function $0$, which is smooth (all of its derivatives are $0$), divergence-free (all its derivatives are $0$), and has derivatives that vanish rapidly (all of its derivatives are $0$). I hope this initial condition is not too difficult to wrap your mind around.

We wish to extend this vector field to a smooth vector field $u : \mathbb{R}^3 \times [0, \infty) \to \mathbb{R}^3$ such that $u(\mathbf{x}, 0) = u_0(\mathbf{x})$. We will also be interested in a scalar field $p : \mathbb{R}^3 \times [0, \infty) \to \mathbb{R}$. The field $u$ gives the velocity of a fluid particle at position $\mathbf{x}$ and time $t$, and the field $p$ gives the pressure at a position $\mathbf{x}$ and time $t$. (We only care about pressure up to an additive constant $c(t)$, because only its spatial gradient is relevant).

The Navier-Stokes equation requires that:

where

for some viscosity parameter $\nu > 0$, and $f(\mathbf{x}, t)$ is a smooth, rapidly decaying external force.

If you unpack the left side of the equation, you find that it gives the acceleration of a fluid particle at position $\mathbf{x}$ and time $t$. The $-\nabla p$ term on the right side gives the acceleration due to pressure differences, the $\nu \Delta u$ term gives the acceleration due to viscosity, and $f$ is an external force.

For some of the subtleties here, one should look at the official Clay problem statement. OpenAI claims specifically that statements C and D have been resolved. In particular, it has (as I understand it) not yet been proven that finite-time blow-up is possible without the external force. But Terence Tao says that the unforced version looks very feasible in light of the work by Alpöge and Buckmaster.

Six days ago, Tao spoke on Mathstodon about the potential implications of an AI-assisted resolution to Navier-Stokes:

A concrete example of how AI advances in solving key open problems could inihibit the future development of a mathematical field can be found in the global regularity problem for the incompressible Navier-Stokes equations (as well as its counterpart for the Euler equations). Until recently, this problem was on track to be one of the most promising examples of an AI-assisted success story, in which AI tools could be used both to solve the problem and to set the stage for the next round of progress in the field. While the problems remain open for now, there is now an increasingly realistic scenario in which a primarily AI-generated solution to the problem appears, but in a fashion that contaminates the problem as a source of further advances.

(…)

But there is now a scenario in which an autonomous AI harness, backed by an enormous amount of computational resources, performs this entire iteration internally, and ends up producing the final ansatz, and thence the solution to the Navier-Stokes regularity problem, while the AI company running the harness keeps the process to arrive at that ansatz almost completely out of public view. Technically, one of the most prominent open problems in mathematics would now be solved; but there would be almost no value added to mathematics as a consequence. It is theoretically possible that with some herculean (and heavily AI-assisted) additional effort by a third party, some portion of the process could be reverse-engineered to recover some actual insight and understanding from the solution; but this would be a far less efficient process than if the solution had been obtained via a diverse combination of both human mathematicians and machine assistance as mentioned above.

Fundamentally, problems in pure mathematics, such as the global regularity problem, serve a different purpose than immediately practical problems, such as that of finding a cure to a specific disease, or increasing the energy efficiency of some engine. In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field through the efforts to solve such problems, and then to digest any partial or complete solutions that emerge for further insights. Prematurely solving the problem by purely AI-powered methods - particularly without full transparency into the solution process - can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.

It seems likely to me at this point that the fate of the Millennium Problems might indeed capture the story of mathematics in the 21st century pretty well. It will consist of a substantial amount of work by humans… followed by AI rapidly chewing through all the rest of it. I roughly saw this coming a decade ago, though I knew not when exactly it would happen. It is now beginning.

My hope is that we are still able to extract insightful new mathematics from the results of AI agents. I do think this should be possible! If nothing else, AI will probably get better at automating the extraction of such insight as we move into the 2030s. But — in a relatively optimistic projection for the future of mathematics — we may be moving from a phase where research programs can center around precisely stated formal conjectures to a phase where we do have to explicitly focus on fuzzier questions, like the development of frameworks that allow humans to have some intuitive handle on the meaning of results.

Or, as Tao put it:

A new proposed competition for AI companies: rather than being the first to announce solutions to unsolved math problems, be the first to announce a new mathematical insight.

Those are the two solved Millennium Problems, though probably there is still some work to do on cleaning up and verifying the Navier-Stokes result, taking care of the unforced version, and so on.

In future posts, we will cover Birch and Swinnerton-Dyer, Hodge, Yang-Mills, Riemann, and P vs. NP. Which will fall next? What will the result of each of them be? Man, I don’t know. But I’m not optimistic about the idea of any of them surviving through the 2030s, that’s for sure — even if we slow down AI development a lot, there’s still plenty of room for frontier models a few tiers above the one that took out Navier-Stokes to be built within the next decade!

Really, I’m not even sure how many of the Millennium Problems will make it to the 2030s.